The volume and topology of a complete Riemannian manifold (Q5933948)

The authors prove the following theorem: Let \(M\) be a complete open manifold with weak bounded geometry (i.e. \(\inf_{x\in M}\text{vol}[B(x,1)]>0\) and \(K_{M}\geq -H^{2}>-\infty \)) and \(\text{Ric}(M)>0\). If for any sequence of points \(\{x_{i}\}\subset M\) and any positive number sequence \(R_{i}\) with \(R_{i}\rightarrow \infty \) holds \(\lim_{i\rightarrow \infty }\frac{\text{vol}[B(x_{i},R_{i})]}{R_{i}^{2}}=0\), then \(M\) is of finite topological type. This theorem partially proves the following conjecture: Let \(M\) be a complete open manifold with weak bounded geometry and \(\text{Ric}(M)\geq 0\). If \(\lim_{i\rightarrow \infty }\frac{ \text{vol}[B(p,r)]}{r}=0\) then \(M\) is of finite topological type. The first part of the paper concerns the proof of the theorem. In the second part the authors discuss some further ideas in order to prove the conjecture presented above, considering an equivalent conjecture. Suppose that \(M\) satisfies \(\text{Ric}(M)\geq 0\), \(\inf_{x\in M}\text{vol}[B(x,1)]>0 \) and \(\alpha _{2}(M)=0\). Then \(M\) has finite topological type provided \( \inf K_{M}>-\infty .\) (\(\alpha _{2}(M)\) is the infimum of \(\varepsilon >0\) such that there exists \(r_{\varepsilon}\geq 1\) and a compact subset \(\Omega \) such that \(\text{vol}[B(x,r_{\varepsilon })]<\varepsilon R_{\varepsilon }^{2}\) for any \(x\in M/\Omega\).) The last part of the paper concerns the study of volume growth. In fact it is proved that the volume growth of a manifold is polynomial. One last comment: this article contains many typing errors which, sometimes, make the text difficult to understand.